In order to really understand what eigenvalues are, one has - of course - to study some math; more precisely linear algebra. I first encountered that in my first year at the University.

Nevertheless: I guess you are familiar with the notion of quantum systems having only a set of certain "allowed" energies. A well-known example is the hydrogen atom, whose lowest energy level is at -13.6 eV, and the next one is at exactly 1/4 that value, -3.4 eV, and so on according to the formula E=-13.6/n^2 eV, where n is any whole number. If you measure the energy of a hydrogen atom you get one of those values and none other. (This also explains the line spectra of gases.)

So in mathematical jargon, we call those energies the "eigenvalues of the Hamiltonian" (or eigenenergies). The word comes from the mathematics we use when solving the equations. I have not heard the word "eigenlevel", but that would be the state that the system is in when it has a definite energy; I would rather say "eigenstate". Think of it as the different orbits that the electron can be in.

Much of the wonders of quantum mechanics comes from the fact that all measurable physical quantities have a set of eigenstates: position, momentum, angular momentum, spin, etc. So if you measure the angular momentum of a particle you are bound to receive one out of a certain set of values, and we can calculate which ones are allowed (given a certain system, e.g. a hydrogen atom or a quantum dot or a neutron star or the Universe ;) ). And the eigenstates of all those quantities are not the same; this actually gives rise to Heisenberg's uncertainty relation.